One of the most fundamental characteristics of imaging optical systems is the ability to produce sharp images of objects and to resolve fine details within these objects. On the other hand, non-imaging optical systems are often required to produce various output light patterns by transforming the propagating radiation. In several photonics applications, for example, it is required to produce high peak irradiance or radiance optical fields with complex, spatially structured radiation patterns.
An optical system's response to a point source, known as the point-spread function (PSF) of the optical system, represents one of the most fundamental characteristics of the optical system. The PSF defines a system's ability to form sharp images or to focus the propagating radiation. The PSF also influences radiance distributions produced by optical systems in the far field. An image of an object produced by an optical system is defined as the convolution of an ideal image with the PSF of the optical system producing the image. The PSF size depends on the radiation wavefront distortions incurred during propagation, including atmospheric effects, optical system aberrations, and diffraction of the radiation as it propagates through the optical system. In the case of optical systems well corrected for aberrations, the shape and size of the PSF is defined only by obscurations and diffraction effects on the system's apertures. Optical systems well corrected for aberrations are termed as “diffraction-limited.” Diffraction effects limit the resolution of optical systems and prevent propagating radiation from being focused into infinitely small spots with infinitely high power densities.
When an object is located at infinity, a diffraction-limited optical system with a circular pupil aperture will produce a focused field distribution known as an Airy distribution in its back focal plane. The Airy distribution consists of a high intensity circular central node surrounded by lower intensity rings caused by diffraction on the pupil aperture. FIG. 1 shows a normalized intensity cross-section of an Airy distribution. The size of the central node of an Airy distribution, referred to as an Airy disk, depends on the wavelength λ of the propagating radiation, the aperture diameter D, and the focal length f of the focusing optics. The Airy disk diameter is defined as:
                              d          Airy                =                              2.44            ⁢                                          λ                ⁢                                                                  ⁢                f                            D                                =                      2.44            ⁢            λ            ⁢                                                  ⁢            N                                              (        1        )            where the ratio N=f/D is referred to in literature as the f-number of an optical system. The Airy disk contains about 84% of the propagating radiation power, while the remaining 16% of the radiation power is distributed between the lower intensity rings of the Airy distribution caused by diffraction of the radiation on the circular system aperture. Equation (1) also describes the central node size of a focused field distribution from a top-hat-shaped collimated laser beam with diameter D and wavelength λ produced in the focal plane of a diffraction-limited focusing optical system with focal length f.
Amplitude transmission masks and phase masks were employed in the past to alter the size and shape of PSFs. FIG. 2 presents changes in the PSF central node diameter, the power outside of the central node, and the power contained in the central node for optical systems with central obscurations produced by opaque, axially-symmetric amplitude masks located at the pupil of the optical system. An increase in pupil obscuration by the amplitude mask leads to a reduction in the output field central node diameter, but at the same time causes an increase in the fractional power diffracted outside of the central node and an associated reduction in the fractional power contained in the central node. Pupil obscurations are produced, for example, by secondary mirrors in reflective telescopes, and result in PSFs with reduced central node widths and an increased amount of radiation diffracted outside of the central nodes.
The idea of using amplitude masks located at the pupil of an optical system to reduce the Airy disk width was first proposed by Toraldo di Francia in 1952. Since then, it has been demonstrated that amplitude and phase masks placed at the pupil of an optical system alter the system's PSF. Several examples of PSF distributions produced with the aid of amplitude and phase masks have been discussed in the past.
The optical path difference (OPD) introduced by phase mask structures is usually chosen to be equal to an odd integer j of half the wavelength 0.5λ of the propagating radiation:OPD=j0.5λ  (2)
In many cases, the lowest integer value j=1 is employed, and the optical path difference introduced by the phase masks equals half the wavelength of the propagating radiation.
The employment of amplitude or phase masks to shape the PSF of an optical system is usually associated with a reduction in the fractional power contained within the PSF central node and the associated increase in fractional power contained outside of the PSF central node. It was previously shown that a reduction in the central node width is associated with a reduction in the fractional amount of power contained within the central node of a focused laser beam and with a respective increase in the fractional amount of power contained within the rings outside of the central node.
FIG. 3 presents the relative changes in PSF cross-sections for an Airy distribution, as well as for an optical system containing a pupil single-step phase mask with four different radial sizes of the phase zone. The ratio of the PSF peak intensity of an optical system employing amplitude or phase masks to the peak intensity of the Airy distribution is known as the Strehl ratio. FIG. 4 presents the calculated Strehl ratios for optical systems with amplitude and phase masks located at the system's pupils as a function of the masks' radial sizes. The figure indicates that the use of amplitude or phase masks to alter the PSF shape leads to reduced Strehl ratios. The reduction in Strehl ratio is, in turn, associated with the reduction in the fraction of radiation contained within the PSF central node and the respective increase in the fraction of the radiation contained outside of the PSF central node.
FIG. 5 presents a circular-shaped uniformly illuminated aperture of a diffraction-limited optical system. FIG. 6 presents the corresponding three-dimensional shape of the Airy PSF intensity distribution in the back focal plane of the diffraction-limited optical system using the aperture from FIG. 5.
Optical systems with central obscurations are widely employed in reflective telescopes and result in PSFs containing diminished fractions of radiation within the PSF central node, as was shown in FIG. 2. FIG. 7 presents a doughnut-shaped uniformly illuminated aperture of an optical system with central obscuration. The radial size of the obscuration shown in FIG. 7 is about 60% of the aperture radius. FIG. 8 presents a three-dimensional shape of the PSF intensity distribution in the back focal plane of the optical system with central obscuration. The PSF of the system consists of a higher peak intensity central node surrounded by lower intensity rings caused by diffraction of the radiation on the doughnut-shaped system aperture. The PSF shown in FIG. 8 has a Strehl ratio of 0.41. The PSF contains 37.3% of the radiation within the central node, while 62.7% of the radiation is diffracted outside of the central node and is contained within the rings of the PSF.
Optical systems with distributed apertures are composed of several spaced apart sub-apertures comprising the system's aperture, and are capable of producing PSF distributions with central node widths smaller than the central node widths of PSFs from individual sub-apertures. FIG. 9 presents the pupil of an optical system containing 6 distributed apertures. The individual apertures of the optical system in FIG. 9 are numbered clock-wise in ascending order. The PSF of an optical system with 6 distributed apertures shown in FIG. 9 consists of a higher peak intensity central node surrounded by a number of secondary lower intensity peaks caused by the diffraction of propagating radiation by the apertures, and is presented in FIG. 10. The PSF of the optical system with distributed apertures has a Strehl ratio of 0.31, and contains only about 20.7% of the propagating radiation within the central node, while 79.3% of the radiation is spread outside of the central node of the PSF. The central node width of the PSF in FIG. 10 is about 5.4 times narrower than the PSF central node widths of the individual sub-apertures comprising the system.
The fractional radiation content within the central node of a PSF produced by an optical system is further reduced if the propagating radiation encounters wavefront distortions. In the case of the optical systems with distributed apertures, the fractional power contained within the PSF central node will be reduced in the presence of wavefront distortions within the individual sub-apertures of the optical system, or when the OPD between the sub-apertures is not equal to an integer number of the radiation wavelength.
FIGS. 11 and 12 present PSFs of the optical system containing 6 distributed sub-apertures in the presence of wavefront distortions producing random OPDs between the individual sub-apertures. The PSF shown in FIG. 11 corresponds to a random set of OPDs ranging from −0.14λ to 0.15λ and listed as OPD set #1 in the second row of Table 1, where λ is the wavelength of the propagating radiation. The PSF in FIG. 11 had a Strehl ratio of 0.22 and contained 17.7% of the total radiation power within the area occupied by the central node of the undistorted PSF. The PSF shown in FIG. 12 corresponds to a second set of phase errors corresponding to random OPD set #2 and ranging from −0.35λ to 0.21λ. OPD set #2 is shown in the third row of Table 1. The presence of these random phase errors between the individual sub-apertures resulted in a field distribution with a Strehl ratio of 0.16 containing only 12.9% of the total radiation power within the area occupied by the central node of the undistorted PSF.
TABLE 1Aperture Number123456OPD Set #1 (λ)−0.110.15−0.060.020.07−0.14OPD Set #2 (λ)−0.190.100.210.17−0.35−0.05
A combination of multiple coherent laser beams into an array, known as an Optical Phased Array (OPA), results in a far field distribution with a central node size significantly smaller than the far field central nodes produced by the individual laser beams. This reduction in the OPA far field central node size is achieved at the penalty of a significant reduction in the fractional radiation power contained within the OPA central node.
The far field distributions are produced at distances L from the OPA that satisfy the far field condition:
  L  >            2      ⁢                        (                      D            OPA                    )                2              λ  where λ is the OPA wavelength, and DOPA is the OPA aperture diameter. Alternatively, the far field distributions can be produced in the focal plane of a lens. OPAs may contain different numbers of individual laser beams and may be arranged into different patterns, with the individual beams taking different sizes and shapes, including Gaussian, super-Gaussian, top-hat, etc. FIG. 13 shows the near field intensity distribution produced by an OPA containing seven Gaussian-shaped coherent laser beams arranged in a circular-symmetric pattern. The individual OPA beams are sequentially labeled 1 through 7, as shown in the figure. FIG. 14 presents the far field irradiance distribution produced by the OPA in the absence of wavefront distortions and phase errors between the laser beams. The far field pattern shown in FIG. 14 contains within the central node about 56% of the total OPA power and has a Strehl ratio of 0.56.
In the presence of wavefront distortions or OPDs between the individual laser beams within the array, the fractional OPA power contained within the central node of the far field is reduced. FIG. 15 presents the far field from the OPA in the presence of random OPDs between the individual beams within the array ranging from −0.35λ to 0.21λ. The specific OPDs associated with the individual OPA laser beams are shown in the second row of Table 2. The far field in FIG. 15 has the peak value reduced to 0.53 of the respective peak value of the far field containing no phase errors. The far field in the presence of the phase errors has a Strehl ratio of 0.30 and contains only 37% of the total radiation power within the central node.
TABLE 2Beam Number1234567OPDs (λ)−0.08−0.190.100.210.17−0.35−0.05